240 6.5  Scanning Probe Microscopy and Force Spectroscopy

could be cut to produce a sharp enough tip). Nonmetallic but electrically conducting carbon

nanotubes have also been utilized, which have some advantages of better manufacturing

reproducibility and mechanical stability. Electrical conduction at the tip is mediated through

the atom of the tip closest to the sample surface, and so the effective radius of curvature is one

to two orders of magnitude smaller than that for AFM tips.

No physical contact is made between tip and sample, and therefore there is a classically

forbidden energy gap across which electrons must quantum tunnel across for electrical con­

duction to occur (Figure 6.8a). In the classical picture, if an electron particle of speed ν and

charge q has kinetic energy E of ~mν²/​2, then it will not be able to travel to a region of space,

which involves crossing a potential energy barrier P of qV where V is the electrical potential

voltage difference if qV > mv²/​2 since this implies an electron with negative kinetic energy in

the barrier itself, and so instead the electron is reflected back from the boundary.

However, in a quantum mechanics model the electron particle is also an electron

wave, which has a finite probability of tunneling through the potential barrier. This can

be demonstrated by solving Schrödinger’s wave equation in the barrier that results in a

nonperiodic evanescent wave solution whose transmission coefficient, T, for a rectangular

shaped barrier of width z, takes the form

(6.30)

T E

z

m

h ^V

E

( ) =

−

−

(

)













exp

2

2

The tunneling electron current IT depends on the tip–​sample separation z and the effective

width wxy in the lateral xy plane of the sample over which tunneling can occur as

(6.31)

I

I

Az

w

T

xy

=

−





0

1 2

exp

/

where

I0 is the equivalent current at zero gap between the tip and the sample

A is a constant equal to (4π/​h)(2m)^1/​2 where h is Plank’s constant and m the free elec­

tron mass

The free electron depends upon the electrical potential energy gap between tip and sample,

and so A is sometimes written in non-​SI units as 1.025 Å eV^−1/​2. The exponential dependence

of IT in terms of tip–​sample distance makes it very difficult to measure the current

experimentally if the distance is greater than a few tenths of a nanometers from a weakly

conducting surface such as a biological sample.

FIGURE 6.8  Other scanning probe microscopies, (a) Scanning tunneling microscopy and

(b) scanning ion conductance microscopy.